Abstract
The aggregate uncertainty is the only known functional for Dempster-Shafer theory that generalizes the Shannon and Hartley measures and satisfies all classical requirements for uncertainty measures, including subadditivity. Although being posed several times in the literature, it is still an open problem whether the aggregate uncertainty is unique under these properties. This paper derives an uncertainty measure based on the theory of hints and shows its equivalence to the pignistic entropy. It does not satisfy subadditivity, but the viewpoint of hints uncovers a weaker version of subadditivity. On the other hand, the pignistic entropy has some crucial advantages over the aggregate uncertainty. i.e. explicitness of the formula and sensitivity to changes in evidence. We observe that neither of the two measures captures the full uncertainty of hints and propose an extension of the pignistic entropy called hints entropy that satisfies all axiomatic requirements, including subadditivity, while preserving the above advantages over the aggregate uncertainty.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Chau, C., Lingras, P., Wong, S.: Upper and lower entropies of belief functions using compatible probability functions. In: Komorowski, J., Raś, Z.W. (eds.) ISMIS 1993. LNCS, vol. 689, pp. 306–315. Springer, Heidelberg (1993)
Dempster, A.P.: A generalization of bayesian inference. J. Royal Stat. Soc. B 30, 205–247 (1968)
Dubois, D., Prade, H.: A note on measures of specificity for fuzzy sets. Int. J. Gen. Systems 10(4), 279–283 (1985)
Harmanec, D.: Toward a characterization of uncertainty measure for the dempster-shafer theory. In: UAI 1995: Proc. of the 11th Conference Annual Conference on Uncertainty in Artificial Intelligence, pp. 255–261 (1995)
Harmanec, D.: Measure of uncertainty and information. In: Imprecise Probability Project (1999)
Harmanec, D., Klir, G.: Measuring total uncertainty in dempster-shafer theory: a novel approach. Int. J. Gen. Systems 22(4), 405–419 (1994)
Harmanec, D., Resconi, G., Klir, G.J., Pan, Y.: On the computation of uncertainty measure in the dempster-shafer theory. Int. J. Gen. Systems 25(2), 153 (1996)
Higashi, M., Klir, G.J.: Measures of uncertainty and information based on possibility distributions. Int. J. Gen. Systems 9(1), 43–58 (1982)
Jousselme, A.-L., Liu, C., Grenier, D., Bossé, E.: Measuring ambiguity in the evidence theory. IEEE Trans. on Systems, Man, and Cybernetics, Part A 36(5), 890–903 (2006)
Klir, G.J.: Uncertainty and Information: Foundations of Generalized Information Theory. John Wiley & Sons, Inc., Binghamton University (2005)
Klir, G.J., Lewis, H.W.: Remarks on ”measuring ambiguity in the evidence theory”. IEEE Trans. on Systems, Man, and Cybernetics, Part A 38(4), 995–999 (2008)
Kohlas, J.: Information Algebras: Generic Structures for Inference. Springer, Heidelberg (2003)
Kohlas, J., Monney, P.-A.: A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence. LNEMS. Springer, Heidelberg (1995)
Kohlas, J., Monney, P.-A.: Statistical Information. Assumption-Based Statistical Inference. Sigma Series in Stochastics, vol. 3. Heldermann (2008)
Maeda, Y., Ichihashi, H.: An uncertainty with monotonicity under the random set inclusion. Int. J. Gen. Systems 21(4), 379 (1993)
Monney, P.-A.: A Mathematical Theory of Arguments for Statistical Evidence. Contributions to Statistics. Physica-Verlag, Heidelberg (2003)
Pouly, M., Kohlas, J.: Generic Inference - A Unifying Theory for Automated Reasoning. John Wiley & Sons, Inc., Chichester (2011)
Schneuwly, C.: Information - eine diskussion. Term Paper, University of Fribourg (1999)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Smets, P., Kennes, R.: The transferable belief model. Artif. Intell. 66(2), 191–234 (1994)
Smith, R.: Generalized Information Theory: Resolving some old Questions and opening some new ones. PhD thesis, University of Binghamton (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pouly, M. (2011). Generalized Information Theory Based on the Theory of Hints. In: Liu, W. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2011. Lecture Notes in Computer Science(), vol 6717. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22152-1_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-22152-1_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22151-4
Online ISBN: 978-3-642-22152-1
eBook Packages: Computer ScienceComputer Science (R0)